Unicyclic signed graphs with maximal energy

TL;DR

This paper identifies the unicyclic signed graphs with the highest energy for various sizes, showing that a specific graph construction maximizes energy except for certain cycles.

Contribution

It determines the maximal energy unicyclic signed graphs for different vertex counts, introducing a specific graph structure as the extremal case.

Findings

$ ext{P}_n^4$ has maximal energy for } n=4,6 ext{ and } n eq 5,7.

The maximal energy graphs are characterized except for cycles $C_5^+$ and $C_7^+$.

The results specify extremal unicyclic signed graphs based on size.

Abstract

Let $x_1, x_2, \dots, x_n$ be the eigenvalues of a signed graph $\Gamma$ of order $n$. The energy of $\Gamma$ is defined as $E(\Gamma)=\sum^{n}_{j=1}|x_j|.$ Let $\mathcal{P}_n^4$ be obtained by connecting a vertex of the negative circle $(C_4,{\overline{\sigma}})$ with a terminal vertex of the path $P_{n-4}$. In this paper, we show that for $n=4,6$ and $n \geq 8,$ $\mathcal{P}_n^4$ has the maximal energy among all connected unicyclic $n$-vertex signed graphs, except the cycles $C_5^+, C_7^+.$